CDNO：面向连续动态物理系统的通道分解神经算子

CDNO: Channel-Decomposed Neural Operator for Continuous Dynamical Physical Systems

摘要: 神经算子为偏微分方程(PDE)的快速近似求解提供了数据驱动路径，但在连续动态系统中仍面临两类关键挑战。其一，现有方法往往难以显式刻画随时间演化的动态模态，导致长时间预测的稳定性与可解释性受限。其二，频率敏感结构在深度传播过程中容易发生混叠并引起高频细节衰减，从而影响物理敏感区域的还原精度。为缓解上述问题，本文提出通道分解神经算子Channel Decomposed Neural Operator (CDNO)，包含两项核心设计。第一，引入极点留数表示法，将算子学习从傅里叶域扩展至拉普拉斯域，以实现显式动态建模。第二，构建通道分解抗混叠架构，通过多分支频率感知通路与受控融合策略增强特征提取的频带分离能力。在涵盖规则网格、结构化网格和点云在内的六项PDE基准测试中，CDNO相对基线方法取得平均15.1%的性能提升，在Burgers方程上提升39.7%，在Navier Stokes方程上提升20.7%。此外，在仅使用80%训练数据的条件下，CDNO即可达到部分基线模型的最优表现，体现出较好的数据效率。进一步的分布外分析表明，现有算子在物理敏感的高频模态上普遍存在明显的性能退化，这提示需要面向高频结构引入更强的物理感知归纳偏置。

Abstract: Neural operators offer a data-driven approach for rapid approximate solving of partial differential equations (PDEs). However, they still confront two key challenges in continuous dynamical systems. First, existing methods often fail to explicitly characterize time-evolving dynamic modes, leading to constrained stability and interpretability in long-term predictions. Second, frequency-sensitive structures are prone to aliasing and high-frequency attenuation during deep propagation, thereby compromising the reconstruction accuracy in physically sensitive regions. To mitigate the aforementioned issues, this paper proposes the Channel-Decomposed Neural Operator (CDNO), which incorporates two core designs. First, we introduce a pole-residue representation that extends operator learning from the Fourier domain to the Laplace domain to enable explicit dynamic modeling. Second, we construct a channel-decomposed anti-aliasing architecture, which enhances the frequency-band separation capability of feature extraction through multi-branch frequency-aware pathways and a controlled fusion strategy. Across six PDE benchmarks covering regular grids, structured meshes, and point clouds, CDNO achieves an average performance improvement of 15.1%, including gains of 39.7% on Burgers’ equation and 20.7% on the Navier-Stokes equations. Moreover, even with only 80% of the training data, CDNO achieves performance comparable to the best baseline models, demonstrating superior data efficiency. Furthermore, out-of-distribution analysis reveals that existing operators commonly exhibit significant performance degradation on physically sensitive high-frequency modes, highlighting the need to introduce stronger physics-aware inductive biases for high-frequency structures.